I've been struggling to understand the explicit details of the completion of an elliptic curve about the origin, and am desperately confused by the explicit details of the resulting group operation. ...

Given the Weierstrass-$\wp$ function,
$$\wp(2x+1+\tau \mid 1, \tau),$$
with half-periods $1$ and $\tau=\omega_2/ \omega_1$, I want to look at the case where $\rm{Re}(\tau) \in \mathbb{Z}$ and I want ...

The charm of elliptic curves is that given one or two integer points, one can find others by the group law. However the easy to guess points from the title just pump me around trough a cyclic group of ...

I'm studying Serre's book "Abelian $\ell$-adic Representations and Elliptic Curves" and in chapter II $\S$2.4 we have this proposition:
Consider $v$ a finite place of $K$ and $F_v \in Gal(K^{ab}/K)$ ...

I was only browsing this book on number theory and the author shows how the solution $(3, 5)$ can be used to generate other exotic rational solutions and then in the end leaves the problem I'm asking ...

I have posted a similar question here.
The GLV method [ref] is used to speed up ECDSA signature generation. In this method, an input scalar $k$ is decomposed into two scalars, $k_1$ and $k_2$. Then ...

Two different circles can have an integer point in common (for example, $P=(1,1)$ belongs to both $x^2+y^2-2=0$ and $x^2+y^2-4(x+y)+6=0$) but any pair of distinct elliptic curves on the class defined ...

I am wondering if there exists somewhere an "animation" of one such map (for some lattice / torus), in the style of the kind of $z \mapsto z^2$ maps one encounters in complex analysis classes (one can ...

Given a prime $p$ such that $3$ does not divide $p-1$, what is the order of the elliptic curve over $\mathbb{F}_p$ given by $E(\mathbb{F}_p)=\{ (x,y) \in \mathbb{F}_p^2 | y^2=x^3+7 \}$
I thought if ...

I was reading Chapter 1 of Elliptic Curves - Number Theory and Cryptography by Lawrence C Washington. He was considering Fermat equations $$a^4+b^4=c^4\text{ and }a^3+b^3=c^3.$$ For the 1st equation, ...

I have found the following formulas for the coordinates of $P+Q$ given that $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ are points on a general curve $y^2 = x^3 + ax + b$ over $\mathbb{R}$:
$$P + Q ...

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...

Cayley–Bacharach theorem: Assume that two cubics $C_1$ and $C_2$ in the projective plane $\mathbb{P}^2$ meet in nine (different) points. Then every cubic that passes through any eight of the points ...

So let $(E,p)$ be an elliptic curve over a field $k$ with a choice of $k$-valued point $p$. Then by Riemann-Roch, there are two global sections of $\mathcal{O}_{E}(2p)$ which gives a double cover of ...

Please help me to solve this question:
Let $H$ be a hyperelliptic curve over $\mathbb{F}_{103}$ given by the equation $ y^2 = x^5+1$. let $J$ be the jacobian of $H$ defined over $\mathbb{F}_{103}$. ...

Consider the elliptic curve
$$E'\colon y^2 = g(x) = x^3 + \frac{1}{432} .$$
One can check that the discriminant of $E'$ is $-1/432$, that $E'$ has complex multiplication, and that $(-1/432)^{1/3} = ...

I was trying to implement the double and addition formulas for elliptic points on a Montgomery elliptic curve. I came across this weird thing which should definitely not be happening.
I took a point ...

Can the discriminant, $4a^3 +27b^2$ of an elliptic curve $$E: y^2=x^3+ax+b$$ be equal to 1.
I believe that this should not be possible otherwise the curve would have good reduction at all primes $p$, ...

Question 1:
If $E/K$ is an elliptic curve defined over a number field $K$, then the Weil Pairing gives me that $K(\mu_n) \subseteq K(E[n](\bar{K}))$. If i identify $Gal(K(\mu_n)/K)$ with a subgroup ...

Let's we have an elliptic curve (EC). Is it possible to construct group $G$ acting on the points of an EC with this property: if $P$ is a rational point on EC then $G(P)$ is also is a rational point? ...

I have another probably very trivial question about elliptic curves. This wikipedia article gives the following formula $|E|+|E^d|=2p+2$ where $E$ is an elliptic curve over $\mathbb{F}_p$ and $E^d$ is ...